Partitioning orthogonal polygons by extension of all edges incident to reflex vertices: lower and upper bounds on the number of pieces

Abstract

Given an orthogonal polygon P, let |∏(P)| be the number of rectangles that result when we partition P by extending the edges incident to reflex vertices towards INT(P). In [4] we have shown that |∏(P)| ≤ 1+r+r², where r is the number of reflex vertices of P. We shall now give sharper bounds both to max_P|∏(P)| and min_P|∏(P)|. Moreover, we characterize the structure of orthogonal polygons in general position for which these new bounds are exact. We also present bounds on the area of grid n-ogons and characterize those having the largest and the smallest area.